Question

Circle A has a radius of `4` and a center of `(5 ,3 )`. Circle B has a radius of `5` and a center of `(1 ,4 )`. If circle B is translated by `<2 ,-1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circles overlap.

Explanation:

What we have to do here is `color(blue)"compare"` the distance (d) between the centres of the circles to the `color(blue)"sum of radii"`

• If sum of radii > d, then circles overlap

• If sum of radii < d, then no overlap

Before calculating d we require to find the 'new' centre of B under the given translation which does not change the shape of the circle only it's position.

`• " Under a translation "((2),(-1))`

`(1,4)to(1+2,4-1)to(3,3)larr" new centre of B"`

To calculate d, use the `color(blue)"distance formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

The 2 points here are (5 ,3) and (3 ,3)

let `(x_1,y_1)=(5,3)" and " (x_2,y_2)=(3,3)`

`d=sqrt((3-5)^2+(3-3)^2)=sqrt4=2`

sum of radii = radius of A + radius of B = 4 + 5 = 9

Since sum of radii > d, then circles overlap
graph{((x-5)^2+(y-3)^2-16)((x-3)^2+(y-3)^2-25)=0 [-20, 20, -10, 10]}